1 Discretisation of PDE's

Much impetus for the development of supercomputing facilities has been to enable calculation of approximate solutions of PDE's, particularly those occuring for large physical systems such as weather, flight or ship hydrodynamics. It is important to realise that there are different levels of approximation which lead to a physical prediction such as drag. The real, physical situation has to be modelled mathematically and that will introduce approximations such as two-dimensionality, semi-infinite domains and so on. Next the resulting differential equations from the model need to be discretised intoducing another set of approximations. Finally the discrete system needs to be solved and again this is often only done approximately, either because physical constants may be known only approximately or because the discrete system has only an approximate numerical solution. You need to be aware that `numerical algorithm' can be used in a number of different contexts and is often used to refer to discretisation of a continuous differential model and to refer to how you obtain an approximate solution of a discrete system.

In describing the variation of physical quantities we use functions: displacement as a function of time and so on. Thus it is important to think about how functions come about. If we consider a simple example, $y=\sin t$, then $y(t)$ could equally well be defined by

(a) geometrically by using a unit circle,

(b) algebraically by using a power series,

(c) as the solution of a differential equation

$$y''+y=0,\ y(0)=0,\ y'(0)=1.$$ (1)

Most of the functions we use in physics and engineering are defined only by the last method: they are defined by a differential equation. Our experience is that most such equations have to be solved numerically.

Another question which should occur to you is why do we have definitions of functions in terms of differential equations in the first place? The main reason is because in engineering and physics we are usually looking at functions which are solutions of conservations laws; laws which relate rates of change of quantities in space and time. Such laws usually fall into only a few simple classes.

Conservation laws

(a) steady problem without sources: what comes into a region must leave.

So if ${\rm\bf q}$ is the flux of some quantity $\phi$, in two dimensions,

$$\nabla .{\rm\bf q}=0.$$ (2)

If then there is a relation such as ${\rm\bf q}=-\kappa\nabla \phi$ for a potential (or temperature or ...) then

$$\nabla (\kappa\nabla\phi)=0$$ (3)

or if $\kappa$ is constant, $\nabla^2\phi =0$, a simple elliptic equation.

(b) unsteady problem: what comes into a volume either stays (and changes the amount in a volume) or leaves.

$${{\partial \phi}\over{\partial t}}+\nabla . {\rm\bf q}=0.$$ (4)

So depending on the relation between the flux and the quantity $\phi$ we get most of the simple PDE's which you have studied in your mathematics courses. As an example, if the flux is determined by diffusion, ${\rm\bf q}=-\kappa\nabla \phi$, then

$${{\partial \phi}\over{\partial t}}=\kappa\nabla^2\phi.$$ (5)

Hence systems of partial differential equations can be just the mathematical expression of conservation laws, and it is very common to end up with two space derivatives ($\nabla^2$ is the divergence of a gradient).

Our plan is to examine how simple differential systems can be discretised (approximate solution of the discretised system will come in section 2) in a number of ways, some of which reflect the physical basis of the equations. We do this via a simple model problem and a series of observations which extend the model problem to cover many aspects of numerical solution of PDE's.